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```THE SILVERMAN-TOEPLITZ THEOREM

by

BRIAN RUDER
B, A,, Port Hays Kansas State College, 1961|.

A MASTER'S REPORT

submitted in partial fulfillment of the

requirements for the degree

MASTER OP SCIENCE

Department of Mathematics

KANSAS STATE UNIVERSITY
Manhattan, Kansas

1966

Approved by:

Major Professor

TABLE OP CONTENTS

Page

INTRODUCTION 1

CESARO SUMMABILITY 6

SILVERMAN-TOEPLITZ THEOREM 21

APPLICATIONS ' 22

ACKNOWLEDGEMENT 29

REFERENCES 30

INTRODUCTION

The modern theory of sumraable series has its origin in the
early eighteenth century. In 1713, a letter written by Leibniz
to Christian Wolf discussing the series

(1) 1 -1 +1-1+ ...,

was published in the Acta Erudltorura. Leibnitz tried to Justify
the value i for the series. He felt that the value i for the
series was reasonable on the basis of his "law of continuity"
and on the basis of the relation:

(2) JL « 1 -x+x^x+ ...,

1+x

which is true for values of x numerically less than unity. How-
ever, Leibnitz felt that the value ^ should be obtainable from
the series (1) without recourse to the series (2). One of his
arguments for the value i was the following: If we take the sum
of an even number of terms of the series (1), the value is al-
ways zero; if we take the sum of an odd number of terms, the
value is always unity. When we pass to the case of an infinite
number of terms, there Is no reason to consider that we have
either an odd or an even number of terms, and therefore no rea-
son for assigning either the value unity or zero to the series,
but rather it is reasonable to assign some Intermediate value.
Moreover, in the process of allowing the number of terms to be-
come Infinite, the values zero and unity for the series occur
with equal frequency. He therefore considered it Justifiable,
on the basis of probability, to assign to the infinite series

a value intermediate between unity and zero, which is precisely
their arithmetic mean, namely -|-.

After this publication of Leibnitz, Euler started work in
this area, and is credited with formalizing the theory of diver-
gent series and suramability methods. Since that time, many
prominent mathematicians have taken an interest in the subject
and have developed the theory to what it is at the present time.

Prior to a discussion of divergent series, suramability
methods, and the Silverman-Toeplitz Theorem, it is essential
that a few definitions be introduced as a means of communica-
tions. In passing, one notes that the theory of convergent
series is contained in the theory of divergent series in that
convergence is nothing more than a logically simple method of
assigning values to infinite series. The fact that there is
more literature written on convergence than divergence is quite
logical since there have been more elementary applications of
convergent series. However this does not in any way detract
from the interest and usefulness of the theory of divergent
series.

Definition 1 _

z

An infinite series Z- a^ is a symbol for a definite

n=0

sequence of complex numbers deducible from it, namely the se-
quence of partial sums. The series

2^ 2^

n»0

" --i-^i^ ^^j^---

has n th partial sums (denoted by 3^) given by

Definition 2

CO

An infinite series 2L a^^ with an n th partial sura S^^,

n=0

is said to be convergent, if lim S^ « S, and we call S the

n-»«>

"sum" of the series.
Definition 3

aoL

An infinite series 2^ a^ with n th partial sum S^^, is
said to diverge definitely to + oo if lim S„ ■ +oo , and is
said to diverge definitely to - oo if lim S_ » - oo

n-*oo *i

Definition h

An infinite series ^ a„ with n th partial sura S„ is
said to diverge indefinitely if it is neither convergent nor
definitely divergent. The series (1), 1 -1 +1 - ..., is an
example of an indefinitely divergent series.

In the study of summability, the only divergent series of
any interest, and the only ones that will be considered in this
paper will be those which are indefinitely divergent.

Definition ^

A summability method is a process which attempts to assign
a number, complex or real, to infinite series.

At this point, a few properties which are desirable in a

summability method will be discussed. The first property which
the method should possess is non-contradiction to the idea of
convergence. (One wants the method to have this property, be-
cause convergent series are of such great value, any summability
method which contradicts it, could scarcely be expected to have
much value.) Therefore one will want the method to satisfy a
permanence condition.

Definition 6

If an infinite series ^ a., converges to S, and if V-

A2 o.-o

( 2^ ajj) «= S-j^, where V is a summability method, we say that the
summability method satisfies the permanence condition. The sec-
ond property that the method should possess is closely related
to the permanence property and is called consistence.

Definition 7

If for every infinite series ^ a„ which converges to S,
we have V ( 2- ^^ ■ S, where V is a summability method, we say
that the summability method V satisfies the consistency property.
If a summability method satisfies both the permanence property
and the consistency property, the following definition is appli-
cable.

Definition 8

A summability method is said to be regular if it satisfies
the permanency property and it also satisfies the consistency
property.

There are other requirements that one might want a

summability method to possess, such as an extension property,
(which means that the summability method sums at least one diver-
gent series) or a compatibility condition. (This property guar-
antees that if an infinite series ^ aj^ is summable by two
different methods, that both methods yield the same value.)
Most methods do satisfy these properties, but in this paper only
the property of regularity will be discussed, so no other prop-
erties will be dealt with in detail.

As should be quite evident from the definition of a regular
summability method, any summability method that is not regular
is of very little practical value. Since regularity is an impor-
tant property, one would like to be able to determine whether an
arbitrary summability method is regular or not. With the aid of
the Silverman-Toeplitz Theorem, one is able to give a partial
solution to this problem. If one is able to put a given summa-
bility method into a form in which the Silverman-Toeplitz Theo-
rem can be applied, then it is possible to determine whether the
method is regular or not. Many summability methods can be put
into this form, so the value of this theorem is evident.

In this paper, the Silverman-Toeplitz Theorem will be proved
and some of the applications of the Theorem will be given.

Before discussing the Theorem, two definitions and examples
of well known summability methods will be given which shall be
referred to throughout the rest of this paper.

CESARO SUMMA3ILITY

Given the infinite series 2^ Uj^, Let Sj^ ■ ^ Uj^.

n«0 k»0

Set C^ - Sq + S^ + S^ + ... + S^.

Then one says that the given series is summable (C,l) to the sura
C if

11m IL « c.

n -♦<» n-fl

This method of summability is due to Ernesto Cesaro. A more
general Cesaro summability method is (C,K) summability, defined
in the following manner:

Sj^^°^ • S^ , and for k^l,

s^(k) .so^^-i) .si(k.i) ^ ... .S^(k-l).

(n » 1,2,3,...)
and one now considers

for each fixed k.

If for some value of k, C^^^^ — » c, one says the sequence [sA
is summable (C,K) to the sum C.

In this paper, only (C,l) summability will be considered as
it lends itself readily to examples.

As an example of the application of the Cesaro method, note

(k) .

3n"''

^n

(Tj

that, with this method of summabllity, the series (1) is surnraable

to the value g, which is consistent with Leibnitz's result.

As a second example consider the problem of finding the

analytic continuation of the infinite series ^ z^ which con-

k-1

1-z^^l .
— and

verges to — for /zj ^1. For this series, S^ » — ~

C.<^+<^4.q4. «;- (1-z)^ + (1-2)^ -h ... + (1-z)"

1-z

g ^ (l-z)(n+l 4- nz -f- (n-l)z + ... + (n-(n-l))z
n "■ *

1-z

Cn

In considering the limit of , one finds that the limit

n+1

exists if and only if jzjil; z / 1. To verify the case where
z « -1, consider the partial sums,
Sq - 1, S^ - 0, S^ - 1, ... ,

^n - Sq + S^ + S^ + ... + S^, where S^ - ^ (1 + (-l)'^).

Therefore, -^IL- . (n^D ^ hU ^ (-1)") . ^ ^ 1 ^ (-D" ^^^
n+1 2(n+l) l+(n+l)

Cn 1

lim —. » ^. It should be noted that this value of i is in

n ■"•** n+1 ■^

accordance with what one would expect to obtain if -1 were sub-
stituted for X in the term , to which the original series

1-x

converged.

8

Before Introducing the second summability method, the rela-
tion between infinite series and their corresponding sequences
will be discussed in a little more detail.

As was mentioned earlier in this paper, to every series
there corresponds a sequence of partial sums. It is also true
that to every infinite sequence there corresponds an infinite
series. The limit of the sequence is the same as the sum of the
infinite series, when either value exists.

Thus, in talking about summability methods, one is Justified

in operating with the sequence of partial sums. The relation of

a series to its sequence was introduced above, so at this point,

only the relation of a sequence to its series will be mentioned.

Given a sequence of complex numbers, (z^, z, , .,,) one can write

this as a series by letting a^, « Zq, a^ - z^ - z., a2 - Zp -

z, , ... ft " z^ - z^ , .... . One then has the series
J- n n n- 1 /

oe oo

n-'O k«l

The reason for introducing the relation of series to sequence is
because the Silverraan-Toeplitz Theorem will be proved in the form
where one considers a summability method as an operator on se-
quences rather than an operator on infinite series. In this case
one can consider a summability method as a transformation, which
takes the given set of partial sums into a new set of partial
sums. At this point the second method of summability, due to
Euler, may be introduced.

An infinite series Uq + u^^ + ... and its sequence Sq,
Sj^, ,., of partial sums, are said to be summable to t by the
Euler transformation (or method of summability) £(r) of order
r, r being a complex constant, if

t^ — ^ t as n — > oo where

'n - t„(r) - X (;:) '•' (i-)"-" s. •

•ies ^ z^ .

Consider this method when applied to the series ^^ z^ . From

k«0
the definition of the transformation, one sees that the new se-
quence, in this case, is given by:

where

k=0 '

1-2

-^ — «s^- s z^

l-z ^ ^
n«0

One can write this transformation as

^ i(n|rMx-r,n-- . J (;;) (r.)Nl-.,

n-k

k-0 k.Q

n n

Now note that

k

2^ (M r^(l-r)^-^ . (r + 1-r)'^, and that J"/")-

k«0

(rz)^ (l-r)"^-^ . (rz + (1-r))", and one can write the

10

transformation as

■'■ (1-z (rz + l-r)'^) .

1-z

If z+l, then the series is summable to its analytic continuation
if and only if | 1-r + rz | ^ 1; this is the same as

C(r) .U^lzllK ^ .
/ r I fr/

If r«0 and r is fixed, the set of values of z for which the
series is summable E(r) consists of the interior of the circle

C(r) with center at the point and radius -^ . As a

r (r(

particular case consider r« — ^, then one has C(~L) « |z + ll;|^l5

for which the sequence is summable and this includes, and is
larger than the original circle of convergence.

From these two examples, one can note that the Euler method
is more powerful than (C,l) summability in the case of the geo-
metric series. However, (C,l) summability is more powerful than
the Euler method when one deals with certain other series.

Up to this point, nothing has been said about the regularity
of these two methods. It will be shown (by making use of the
Silverman-Toeplitz Theorem) that (C,l) summability is regular
and that E(r) is regular if and only if certain restrictions are
placed on r.

The following Infinite matrices will be used as reference
matrices throughout the rest of this paper.

11

u • s

^00 ^01 ^02
^10 *11 ^12
^20 ^21 ^22

^nO ^nl ^n2

n

n

T

U is an infinite matrix denoted by (a^^). S and T are infinite
column matrices. The product U • S « T expresses a method of
transforming the sequence [sj to the sequence (t |. In the
problems to be discussed, the S matrix will have as elements the
partial sums of the infinite series under consideration.

The method of proof selected for this paper requires that
a few lemmas be proven prior to consideration of the Silverman-
Toeplitz Theorem.

Lemma 1

eo

1

If the series ^ p^^q^ is convergent for every bounded
n-0

es

or merely for every convergent sequence Jpn{ , then the seri

y

-C_ qn is absolutely convergent. It is convenient, at this
n«0

place to introduce the function signum, abbreviated sgn, which
is defined as follows:

12

/i|i if z it

sgn z

■ 1 if z « .

Proof.

If the sequence [vn] ^^ required to be bounded, then let
Pjj » sgnq^ (which is obviously bovmded) and one has

n=0 n=0 n=0

which is absolutely convergent and the first part of the lemma

is proved. If the sequence jPn? i^ convergent, the following

argument can be given. Assume ^ /QhI ^^ divergent. Let

n»0
n

^n " Z-. /qyl , then as n — > a? approaches Infinity
v=0 "

monotonically. The series ^~ -^ is divergent, for by

n«o ^n

writing

n+s n+s

v=n v«n

one notes that
n+3

v»n ^ Ti+s Si+s

13

for s sufficiently large. Now set p « t—J and it is true

^n*^n
Pj^ -» as n gets large since Q^ approaches infinity. Now by

is ^

hypothesis ,^ Pn^^n converges, but
n«0

y p „ . f JM . , . 2 I2nl

which leads to a contradiction and therefore our assumption must

00

have been wrong and Z-Knl ^^ convergent and the lemma is
proved. Lemma 1 shows that any condition on the U matrix given
above which insures the existence of the T matrix for all bounded
or convergent sequences [Sj^j, will have to include the condition:
the sum of the absolute values of the elements of each row of U
is finite, that is

"*a
nv

v«0

exists for each n. As a matter of convenience, let V « / a

» n ^ '

v»0

and W^ - ^ I &^^l for the rest of this paper.
v«0

Lemma 2

A necessary and sufficient condition that the sequence /"t ?
be bounded whenever the sequence fsl is bounded is that the
sequence jw^? be bounded.

11^

Proof.

Consider the sufficiency first (that is if fSy^f is bounded
and jWj^f is bounded, then JT^f is bounded.) Let M be the least
upper bound (abbreviated l.u.b.) of [Sj^), and let N be the l.u.b,
of Wj^, then JT^^j ^ M-N and [t^J is bounded.

For the necessity of the condition, note first of all that
the elements of each column of U must be bounded. To show that
this is true consider the set of sequences S « ^nv> ^ * 0»1»2,
... where ^^^ is the Kronecker delta defined as follows:

fl if n « V
"^ (o oti^erwise.

Then one has T^ « a^^, n « 0,1,2, ••• and this sequence must be
bounded for each v according to the hypothesis of the lemma.

Now if Wj^ is unbounded, a bounded sequence yS^ f will be
constructed for which {Tj^T is unbounded. Without loss of gen-
erality, it may be assumed that W^^ tends monotonically to in-
finity, for is this were not true, a subsequence ^Xj^T of W^^ for
which this were true could be selected. Now define the sets of
integers (u^), (k^^), (y^^) as follows: Uq - 0; k^ = 0; Jq is the
smallest integer greater than k^ + 1 such that ^ /a /^ 1.

v«yo
The reason that one can do this follows from lemma 1, that is if
the T matrix is to exist (and this is the only case to be con-

sidered here then ^ l ^r,^rl must exist for all n and in

v«0

15

particular for n » 0. If Z^ / ^nv / ©^Ists, then for v large
enough, it is true that 2__ I *ov / ^ ■'■• ^°^ suppose that the

first m of each of these sets, (u^), (k^), and (y^^) have been
chosen; choose u^, k^, y^, as follows. First choose u^ as the

smallest integer greater than u . for which there are numbers

m-i

k, greater than J^j^^i such that

k
^^^ 2 /^uv/ ^ ~W^ - 1 for u - u^ .

Now Uj^ exists since W^^ tends monotonically to infinity and the

elements of each column of U are bounded, ie, la I ^ M(v)

' I nv ' ' ' f

n « 0,1,2,..., V = 0,1,2,.,. , This means that once u__t has
been chosen, one can select v large enough so that a Uj^ exists
which satisfies (a) because the columns of U are bounded. Let

^m ■ ^m-l "•■ ^' ^m ^^ ^^®" ^ ^ ^°^ which (a) is true. Take y^^
as the smallest integer greater than k^^ + 1 such that

CO

for u « u_ ,
m

v=y

m

This is possible by lemma 1 as was shown above. The sets (u^),
(k^), and (y^) are completely defined and one has

^O^yQ^^l'^yi-^k^^y^-^k^-ry^ ^ ... .
Now define the bounded sequence 7 S^ j for which JT ? will be

16

iinbounded, as

otherwise ^yn-l-^~^n^*
The sequence j S f is obviously bounded and one can see for

-i

V"0 ' / v=«0 *Siv ' v«0

Now ^0_ yj^_^

. ... J.

v«0 v«k.

^ yn-1

n+1

^n «^

Z

v«0 ^"yn

^n ^^

wher-e | "u 1 ^ Z / ^uv / * 2- j\J-

In defining j^^) many zeros were introduced and this is the

reason for the strict inequality. Now to proceed, one has

^n yn-1 ^

yn-1 kn 02-

^"k^+1 v»0 v»y

n

Note that R is positive so that R > - R. Now replace R by - R
in (1) and then use (2) to get

17

CO ^n-l

v«=0 v=k+l

k oo

and W^ tends monotonically to infinity. The last inequality was
obtained by using (a) and (b) , that is

00

-2 Z l^v|--2-l--2.

v=yn

Since the sequence l^rx) ^^ bounded, the sequence Iw ? is un-
bounded, and the sequence Jt^J is unbounded, one can see that
the condition that | W^ V be bounded is necessary, and lemma 2
is proved.

Lemma 3

A necessary and sufficient condition that the sequence fT^?
has a limit whenever the sequence |*S^? has the limit zero is

that - ^

(a) 7^n) ^® bounded, i,e. W^ ^ W and.

16

(b) lim a^^^ = ly exists for each v.
n —♦017

.tof T^» Z_ 1,

Then the limit of '^xi' Z— \ \ '

Proof.

Consider the sufficiency first.

ex>_ k o»

'^n ' 2i, ^nv \ ' Z. ^nv ^v + 2- ^nv ^y and
v=0 v-0 v=k+l

k oo

K - Z ^nv Sv ^ Z l^uvl (S^l - lie W, ^^«^«
J v=0 / v»k+l

Lj^ ■ l.u.b. jS^j . Note that since the sequence Is (has
limit 0, Ljj tends monotonically to 0, Now

Fn- Z ^nv \j ^ Lj^W , so that

k
- ^ ^ - "^n - Z ^nv ^ ^ ^ W » ^° t^a*

k k

nv V k n Z nv v ic

Let n approach infinity, and one has

k k

Z ly S^ - Lj^ W :^ lim_T^ - ill T^ ^ Z 1^ S^ + \ W.
v-0 v-0

19

The sequence j T^V being bounded by lemma 2, these inequalities
for k approaching infinity yield

v»0 v=0

1 S :^ lim T^ ^ lim T^ ^ ^ 1 S„
V V n n Z— V v

since L^^ — > 0, This proves the existence of lim T_, the con-

n "

Zi

vergence of the series Z- 1^ S and the equation lim T,

•o

VM V V " wvi-«w^w*^ a.^m *j^

v=0

To show the necessity of the conditions, a method similar

to the one used in lemma 2 will be employed. To show that the

existence of lim a „ is a necessary condition, consider the set
n *

of sequences S^ « S^^ (Kronecker delta) n = 0,1,2,... . Then
one has T^ - a^^^ and lim T^ exists by hypothesis. To show that
the numbers W^^ must be bounded, consider the set of integers
defined in lemma 2, but now choose with the given kj^, yj^, u
^ " 0,1,2,... ^sqn auv

\

K

^n ^ V "^ ynJ « " "n

otherwise.
S^ is defined in this manner because the sequence Ss ? now has
limit (since the sequence { W^ ^ is unbounded) and this is part
of the hypothesis. The procedure is now exactly the same as in

lemma 2 and for u « u_

l-ul^iRT.

20

which becomes infinite as n — >oo , and the proof is complete.

Lemma Ij.

A necessary and sufficient condition that the sequence
I'^^i has the limit zero whenever the sequence /Sj^( has the
limit zero is that

(a) the sequence l^^] is bounded and,

(b) lim a «0 v"012 '^••»

Proof.

The proof follows almost directly from lemma 3, From lemma
3 it is seen that

CO

li'" "^n ' Z. W » ^^«^« li'n ^nv ' \ *
v*0

if lim Q.^^ - one has lim ^^ « 0. If lira ^^ = 0, then one can
show lim a^^^ « in the following manner. Consider the sequence

S^ « (0,0, 0,0, ...0,1,0,...) where the 1 is in the i th

position. Then lim S. « 0. This says T^ « a . u - u but

•^ i ui * n

lira T^ - « 1^. The rest of the proof follows from lemma 3.

Theorem 1

A necessary and sufficient condition that the sequence

Un) ^^ ^ limit whenever the sequence Js^^j has a lirait is
that

(a) I W^^ is bounded,

(b) n iilU a^^. 1^ V - 0,1,2,...,

21

oo
ir,) 11m \/ „ lim /

Ej^y exists

v«0

Proof.

If one applies lemma 3 to the sequence ? S " S f where

S » 11m S^ then one has

oo

Za (S - S) « ^^^ (T - S V )
nv^ V ' n ^ n n' »

v-0

CO

so that IJ^ra T^ - s lira V^^ + ^L V^v ' ^^

v«0

and by condition (c) 11m V^ exists, so that If the sequence /s ?
has a limit, T^^ also has a limit which Is given by

11m T^ - S 11m V^.

SILVERMAN-TOEPLITZ THEOREM

Theorem 2

A necessary and sufficient condition that the sequence )T T
has a limit whenever the sequence 5Sjj? has a limit and that the
two limits are equal Is that

(a) 1^7 is bounded,

<^) \ " ni^So ^nv -0, V - 0,1,2,...

^<5> nli« V„ - 1.
n — ioo n

22

Proof.

Prom Theorem 1, one can see that condition (a) is a neces-
sary and sufficient to insure that )t I has a limit whenever
^Sj^ V has a limit. Also from Theorem 1, recall that

lim Tjj - S lim Vj^ + 2_ 1^ ^\ " S) .

v«0

If Iv -n'^'i ^nv " °' ^"^ ^^ I^'" ^n " ^ ^^^"^ ^^"^ "^n " ^
If lim Tj^ « S then lim T^^ •= S - S lim V^^ which says lim V^ - 1
The necessity of condition (b) follows from lemma \\.^ and the
proof is complete,

APPLICATIONS

As the first application of this theorem, If will be shown
that the (C,l) summability method is a regular method. Recall
that the summability method was defined by

lim _JL- where C^ - S^ + S, + ... S„.
n — *oo n+1 n 1 n*

Therefore, one has as the infinite matrix of transformation and
the transformed set of sequences the following:

23

10

^i-000 ..

2^ ^ L
3 3 3

_1 1 1_

n+1 n+1 n+1

n

%

So + Si

2

So + Si +

sz

3

So

+ s, + ...

^n

n+1

•

•

•

Now consider the first condition of the theorem: j^Yii
bounded. In this case

Is

CO

VcQ
and condition (a) is obviously satisfied. Condition (b) is:

1^ " ^^ Q-xiy - 0; V » 0,1,2,'..; for an arbitrary v one has

^nv ■ ;rrr ^^^ ^^^ -^ " O- condition (c) is: 11m V„ « 1.
nv n+1 n-^oo n+1 n->oo n

In this case one has

lim V.

CO

n " lira JL.

v«0

nv

lim

I i

v-1

lim n (^) . 1.

2k

From the fact that all three conditions of the Silverman-Toeplitz
Theorem are satisfied, one knows that (C,l) summability is a
regular method.

As a second example, consider the Euler method of summa-
bility. Recall that the method is defined for a fixed complex
number r. Using the Silverman-Toeplitz Theorem, the values of
r for which this method is regular shall now be determined. The

Euler summability method is given by:

n

^n " X ^nk Sk ,
k-0

where

so that

CO

Or + (l-r))'' - "^ a^

k«l

^nk

(2 J r^ (l-r)"-k,

The infinite matrix of transformation is given by:

1
1-r

r

(l-r)*^ 2r(l-r)

/, »n n /_ \n-l
(l-r) ^ r(l-r)

. . .

n 3, «n-s ^
g r (l-r)

To determine the set of r's for which the Euler method is regular

25

assume that the first condition of the Silverman-Toeplitz Theorem
is satisfied, that is ) W^^ ( is bounded. From this it is seen

CO *- ^

that Z— I a^^^ / is boxinded. For the Euler transformation, one

v=0

m

sees that 2_. j ^nv\ " U^l + |l-r( )"^, n - 0,1,2,... .
v=0

In the case where r is an arbitrary complex constant, let

r = a + bi and consider r «= Re^* where R = Va^ + b^ and

© = arc tan — . Then
a

(/r| + |l-r|) = (R + 1 -R (cose + isine ) )
- (R +^1 - 2Rcose + R^)

(R + Vl - 2R + R^ + 2R - 2Rcos© )

R . (1-R) \/l^ 2^il^so2 ,
» (1-R)2

Since 1 - cos© 2 0, \l + ^^(^'^ose ) ^ ^^ ^^^

y (1-R)2

(|r| + (l-r|) > R + (1-R) .1-1,

and equals 1 if and only if © = 2k7r ; k « 0,1,2,... . If

©^ 2krr , then { \r] + /l-r/) > 1, and one can readily see that
condition (a) is not satisfied since

(Irj + |l-r/) -» 00 as n -^oo .

Therefore, if condition (a) of t he theorem is to be satisfied

26

one must have 9 = 2k 7J, and this says that r must be real. If
r is real then condition (a) is satisfied if and only if
^ r £ 1.

After consideration of the first condition, r has been re-
stricted to be real and in the interval ^ r 6 1, if the Euler
method of summability is to be regular. The two cases where
r = and r = 1 will be considered separately. For these values
the infinite matrices of transformation are given by:

10 0...
10 0. . .

10 0..

• • • • • « •

10
10
10

0' • '010

n

'n

'n

27

The first set of matrices are those obtained for r = 0, and the
second for the case where r « 1. In the first case, condition
(b) of the Silverman-Toeplitz Theorem is not satisfied for v » 0,
and therefore when r >= 0, the Euler method is not regular. In
the second case where r = 1, one can see that this is Just the
identity transformation, which is obviously regular.

For the last two conditions of the theorem, one needs only
to consider r real and in the interval ^ r < 1. Consider con-
dition (b) ni^^ a^v •= 0, for v = 0,1,2,... . To show that this
condition is satisfied, consider the infinite series whose nth
term is the a^^^ of condition (b) . It will be shown that the
series converges and therefore that the nth term approaches 0,
which in turn will show that ^^^^ a^^ - 0. Consider the in-

finite series

I

n+v , „ ,_ .n-v

rv (i-r)'

where v is fixed. Now apply the ratio test to this, and one has

lim
n -*»

f

n+l+k) . ,n+l-k

w

(l-r)

n-k

lim
n -><»

1 +

n^ H

= )-/•

For <«i r ^ 1, this series converges and therefore the nth term

approaches and ^^^ a =0.

n — \$» nv •

Condition (c) is obviously satisfied since it has been shown

28

that

n

v=0

but since r>0 this is { r + l-r)" «= (1)" = 1. The conclusions
that one can draw from the Silverman-Toeplitz Theorem concerning
the Euler method of summability is that the method is regular if
and only if r is real and in the interval < r ^ 1.

Prom these two examples, the importance of the Silverman-
Toeplitz Theorem as a tool in the theory of divergent series is
evident. It can be used to determine whether a given summability
method is regular. In certain cases, It can be used to determine
the values for which a method is regular. In either case, the
Theorem gives much information and also leads to many interesting
mathematical problems.

ACKNOWLEDGEMENT

The writer wishes to express his sincere appreciation to
Dr. C, P. Koch for introducing him to this subject and for his
patient guidance and supervision given during the preparation
of this report.

REFERENCES

Kaplan, Wilfred.

PP. 350-375.

Theory and Applications of Infinite Series. New York:
Hafner Publishing Co., 1928, pp. kSl-k^S.

Moore, Charles N.

Summable Series and Convergent Factors. New York: American
Mathematical Society Colloquium Publications, Volume 22, 1938,
pp. 1-10.

Szasz, Otto.

Introduction to the Theory of Divergent Series. Stechert-
Hafner, Inc., 1914^, pp. 1-29.

THE SILVERMAN-TOEPLITZ THEOREM

by

BRIAN RUDER
B, A., Port Hays Kansas State College, I96I4.

AN ABSTRACT OP A MASTER'S REPORT

submitted in partial fulfillment of the

requirements for the degree

MSTER OP SCIENCE

Department of Mathematics

KANSAS STATE UNIVERSITY
Manhattan, Kansas

1966

The modern theory of summability inethods and divergent
series had its start in the eighteenth century. Some of the
first men to work in this area were Leibnitz and Euler.

In this paper methods for assigning values to certain diver-
gent series are discussed. If these methods are to be of any
value, certain restrictions must be placed on them. One of the
more Important conditions that the method must possess is called
regularity. To say that a summability method is regular means
that the method "sums" convergent series to the value that the
series converges to in its circle of convergence. The Silverraan-
Toeplitz Theorem supplies three necessary and sufficient condi-
tions that a method must satisfy if it is to be regular. A
proof of the Silverman-Toeplitz Theorem is given in this paper.
After the Theorem is proved, two applications of the Theorem are
given.

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